How to Find Nash Equilibria — Best Response Method

Use this recipe to find all pure-strategy Nash Equilibria in a small Normal-Form Game. The underlining (or "doubly-circled") method is mechanical, fast, and the standard exam approach.

1. For each column, find the row that gives Player 1 the highest payoff → underline that payoff.
2. For each row, find the column that gives Player 2 the highest payoff → underline that payoff.
3. Any cell where both payoffs are underlined is a Nash Equilibrium.

Common pitfalls

• Confusing the first and second payoffs in each cell. By convention the first number is Player 1's payoff (row chooser), the second is Player 2's (column chooser).
• Stopping at the first NE. Games can have 0, 1, or multiple pure-strategy NE — finish the full underlining pass before reporting.
• Forgetting the mixed-strategy NE. If the underlining produces zero pure NE (or you've been asked for all equilibria), set up an indifference condition and solve for the Mixed Strategy equilibrium.
• Treating ties carelessly: if a row and column tie at the same payoff, both are underlined. Do not silently break the tie.

Worked example

Prisoner's Dilemma (marketing game):

P1's best response is "Hire" against both columns (6 > 5 and 4 > 3) — underline 6 and 4 in the Hire row. P2's best response is "Hire" against both rows — underline 6 and 4 in the Hire column. The only doubly underlined cell is (Hire, Hire) → (4, 4), the unique Nash Equilibrium.